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Theorem aev 2036
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 1930. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2101, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.)
Assertion
Ref Expression
aev  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Distinct variable group:    x, y

Proof of Theorem aev
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 aevlem1 2032 . . 3  |-  ( A. x  x  =  y  ->  A. u  u  =  w )
2 ax6ev 1817 . . . 4  |-  E. u  u  =  v
3 ax7 1870 . . . . 5  |-  ( u  =  w  ->  (
u  =  v  ->  w  =  v )
)
43aleximi 1714 . . . 4  |-  ( A. u  u  =  w  ->  ( E. u  u  =  v  ->  E. u  w  =  v )
)
52, 4mpi 20 . . 3  |-  ( A. u  u  =  w  ->  E. u  w  =  v )
6 ax5e 1770 . . 3  |-  ( E. u  w  =  v  ->  w  =  v )
71, 5, 63syl 18 . 2  |-  ( A. x  x  =  y  ->  w  =  v )
8 axc16g 2033 . 2  |-  ( A. x  x  =  y  ->  ( w  =  v  ->  A. z  w  =  v ) )
97, 8mpd 15 1  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1452   E.wex 1673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674
This theorem is referenced by:  axc16nf  2037  axc16gALT  2206  2ax6e  2289  wl-naev  31906  wl-ax11-lem2  31960
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